I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse. By 1800, they are almost non-existant. Many B/M/D events either weren't recorded, or haven't survived.
For a James Byers and Ann ? marriage in 1884 (exact), Co. Cavan, I found 1 record. However, the parents' names don't match your info. Note - Mandatory B/M/D civil registration began in 1863. These records are kept by the General Register Office. Query it at WWW.GROIRELAND.IE for copies of the marriage cert. The cert should give you more info.
For a Robert Byers and Ann/Jane McQuade marriage in 1850 +/- 5 yrs, anywhere in Ireland, I found 0 records. Note - Protestant B/M/D records are the least likely to survive.